Journal of Theoretical Biology 395 (2016) 103–114

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Journal of Theoretical Biology

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Adaptive threshold harvesting and the suppression of transients

Juan Segura a, Frank M. Hilker b, Daniel Franco a,n

a Departamento de Matemática Aplicada, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia (UNED), c/ Juan del Rosal 12, 28040,Madrid, Spainb Institute of Environmental Systems Research, School of Mathematics/Computer Science, Osnabrück University, Barbarastr. 12, 49076 Osnabrück, Germany

H I G H L I G H T S

� We present an adaptive harvesting strategy to reduce population fluctuations.

� The strategy is useful for exploited species or to prevent outbreaks.� It provides an alternative to adaptive limiter control.� Transient dynamics can profoundly raise the short-term yield.� We propose adjusted control strategies that reduce the length of transients.

a r t i c l e i n f o

Article history:Received 27 August 2015Received in revised form28 January 2016Accepted 30 January 2016Available online 6 February 2016

Keywords:Population size fluctuationsStabilizationAdaptive limiter controlTransient cancellationShort-term yield

x.doi.org/10.1016/j.jtbi.2016.01.03993/& 2016 Elsevier Ltd. All rights reserved.

esponding author.ail address: dfranco@ind.uned.es (D. Franco).

a b s t r a c t

Fluctuations in population size are in many cases undesirable, as they can induce outbreaks andextinctions or impede the optimal management of populations. We propose the strategy of adaptivethreshold harvesting (ATH) to control fluctuations in population size. In this strategy, the population isharvested whenever population size has grown beyond a certain proportion in comparison to the pre-vious generation. Taking such population increases into account, ATH intervenes also at smaller popu-lation sizes than the strategy of threshold harvesting. Moreover, ATH is the harvesting version of adaptivelimiter control (ALC) that has recently been shown to stabilize population oscillations in both experi-ments and theoretical studies. We find that ATH has similar stabilization properties as ALC and thusoffers itself as a harvesting alternative for the control of pests, exploitation of biological resources, orwhen restocking interventions required from ALC are unfeasible. We present numerical simulations ofATH to illustrate its performance in the presence of noise, lattice effect, and Allee effect. In addition, wepropose an adjustment to both ATH and ALC that restricts interventions when control seems unneces-sary, i.e. when population size is too small or too large, respectively. This adjustment cancels prolongedtransients.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Due to their recurring and sometimes unpredictable ups anddowns, fluctuations in population size pose several challenges forbiological conservation and the management of wildlife andexploited populations. Intervention strategies have been devel-oped to reduce the outbreak frequency and extinction probability(Hilker and Westerhoff, 2007a), stabilize the fluctuations (Hudsonet al., 1998; Korpimäki and Norrdahl, 1998; Desharnais et al.,2001), and maximize the yield of harvested populations (Landeet al., 1995; Hudson and Dobson, 2001).

Here, we propose and analyze a harvesting strategy that takeseffect only if the population size has grown by at least a certainfactor in comparison to the previous census. This conditionalstrategy differs from textbook strategies like constant-effort andconstant-yield harvesting, as it responds only to populationincreases sufficiently large. We shall refer to this strategy asadaptive threshold harvesting (ATH) and the harvesting version ofadaptive limiter control (h-ALC), because it is closely related tothreshold harvesting, also known as limiter control (LC), on theone hand and to adaptive limiter control (ALC) on the other hand.

Threshold harvesting removes individuals from a populationwhenever population size exceeds a fixed threshold value (Landeet al., 1997; Fryxell et al., 2005). This harvest control rule isequivalent to limiter control, which is a method originating from

J. Segura et al. / Journal of Theoretical Biology 395 (2016) 103–114104

physics (Sinha, 1994; Corron et al., 2000; Wagner and Stoop, 2001)and has been applied to problems as diverse as computer archi-tecture design (Ditto and Sinha, 2015), cardiac rhythms (Glass andZeng, 1994), commodity markets (He and Westerhoff, 2005), andpopulation dynamics (Hilker and Westerhoff, 2005, 2006). Limitercontrol methods have the advantage that no detailed informationof the system is required, which is why they are easy and fast toimplement. Similar to some other population control methods(e.g. McCallum, 1992; Parthasarathy and Sinha, 1995; Solé et al.,1999; Liz, 2010; Dattani et al., 2011; Franco and Perán, 2013; Tunget al., 2014), they directly affect the state variables, i.e. the popu-lation size, by restocking (adding) or harvesting (removing) indi-viduals. This approach seems particularly apt for ecological sys-tems that are characterized by intrinsic uncertainty and ‘inacces-sibility’ of parameters such as life-history traits.

Adaptive limiter control has been recently proposed by Sahet al. (2013). The idea of ALC is to add individuals to the populationwhenever the population size falls below a certain fraction of itsvalue in the previous generation. The term ‘adaptive’ follows fromthe fact that the threshold value of the population size triggeringcontrol is a fraction of the previous population size, and as suchvariable over time. The efficacy of ALC to stabilize biologicalpopulations has been shown in laboratory experiments on popu-lations and metapopulations of the fruit fly Drosophila melanoga-ster (Sah et al., 2013) as well as by analytical results (Franco andHilker, 2013) and numerical simulations (Sah et al., 2013; Francoand Hilker, 2013; Tung et al., 2014; Sah and Dey, 2014). ALC istherefore one of the few well-studied control strategies in ecology.

As a restocking strategy, ALC is likely to be applied in thecontext of conservation and re-introduction programs. Dependingon the management objective and the ecological objective, how-ever, there is clearly a need for an alternative based on harvesting.For instance, management programs directed to the control of pestspecies and species of commercial value (e.g., fisheries) aim toremove individuals from the population. Obviously, ATH appearsrelevant in the context of controlling outbreaks of nuisance speciesand exploiting biological resources.

Moreover, independent of any management objectives, theremay be logistical restrictions of restocking strategies. They requirethe availability of a versatile source of individuals for supple-menting the population if needed. However, such a pool of indi-viduals may be difficult or even impossible to create or to maintainin practice. For instance, some organisms cannot be kept in cap-tivity or do not reproduce in such conditions. Furthermore, theremay be issues related to translocation and releasing individuals.Maintaining and managing a stock may be costly, labor-intensive,logistically challenging, and time-consuming. By contrast, remov-ing individuals is certainly ‘easier’ than restocking in some situa-tions. It should be noted, though, that removal methods may bedifficult to implement as well and may raise ethical concernsabout killing animals.

Being based on harvesting rather than restocking, the harvestcontrol strategy presented in this paper is the harvesting version ofadaptive limiter control. That is, whenever the population sizeexceeds a limit that is a certain proportion of the population size inthe preceding generation, harvesting takes place and reduces thepopulation size to that limit. In contrast to threshold harvesting, thecritical population size above which interventions take place is notfixed, but is adaptive in response to the previous population size.

While the kind of control (restocking vs. harvesting) is moti-vated by the biological application, replacing restocking by har-vesting may appear straightforward from a mathematical point ofview. However, there is no reason to believe that the dynamicalbehavior induced by ALC on the one hand and by its harvestingversion ATH on the other hand is similar. This becomes clear whenconsidering other control strategies that have harvesting and

restocking variants. For instance, proportional feedback control(Güémez and Matías, 1993) is able to stabilize a populationtowards a positive equilibrium when a constant proportion of thepopulation is harvested (Liz, 2010), but the restocking variantadding a constant proportion of the population fails to stabilizethe equilibrium (Carmona and Franco, 2006). Constant feedbackcontrol provides another dramatic example. While adding orremoving a constant number of individuals each generation canstabilize chaotic dynamics (McCallum, 1992; Gueron, 1998; Stoneand Hart, 1999), the latter form of intervention can drive thepopulation extinct at small removal rates, even when the popu-lation is able to persist for higher removal rates (Sinha and Par-thasarathy, 1996; Schreiber, 2001). These examples arise in thesimplest case of single-species models given by unimodal mapsthat we also consider in this paper.

In the next section, we introduce the mathematical modeldescribing adaptive threshold harvesting. We then analyze itseffect on the constancy stability in terms of two different mea-sures, namely the fluctuation range and the fluctuation index.Mathematical proofs of the results presented in this section can befound in the Appendix. Since ATH is a harvesting strategy and theindividuals removed may be actually of economic interest (e.g. infisheries), Section 4 considers the mean yield per generation, bothin the long-run and the short-run. Section 5 focuses on the short-term dynamics generated by the interventions. Transients areoften ignored in theoretical studies of ecological systems, which iswhy we discuss in some detail how to deal with them. In parti-cular, we propose adjusted versions of both ATH and ALC thatreduce the length of transients and thus accelerate the approach tothe long-term dynamics. Section 6 explores the robustness of ATHagainst noise, lattice effect, and Allee effect.

2. Adaptive threshold harvesting

2.1. Population growth model

The effect of a control method on a biological populationdepends on the underlying population dynamics, so we start bydescribing the model of the uncontrolled system. We assume thatthe population dynamics is described by a first-order one-dimensional difference equation

xtþ1 ¼ f ðxtÞ; x0A ½0;1Þ; tAN; ð1Þwhere xt denotes the population size at time step t. The populationproduction map f is assumed to satisfy the following conditions:

(C1) f : ½0;b�-½0; b� (b¼1 is allowed) is continuously differenti-able and such that f ð0Þ ¼ 0, f ðxÞ40 for all xA ð0; bÞ andf 0ð0þ Þ; f 0ðb� ÞAR.

(C2) f has two non-negative fixed points x¼0 and x¼ K40, withf ðxÞ4x for 0oxoK and f ðxÞox for x4K .

(C3) f has a unique critical point dAð0;KÞ in such a way thatf ðdÞrb, f 0ðxÞ40 for all xA ð0; dÞ and f 0ðxÞo0 for all xAðd; bÞ.

These conditions describe a unimodal population productionfunction peaking at x¼d and are standard assumptions in thestudy of discrete-time population dynamics (e.g. May, 1976; Singer,1978; Cull, 1981; Schreiber, 2001; Carmona and Franco, 2006;Franco and Hilker, 2013). Biologically speaking, the population hastwo fixed points, namely (i) the extinction state x¼0 and (ii) apositive equilibrium x¼K, which corresponds to the carryingcapacity of the population. Initial population sizes are smallerthan b, and the dynamics are overcompensatory with no demo-graphic Allee effect. Examples include the frequently consideredpopulation dynamics models in their overcompensatory regimes,

Fig. 1. During the first 20 generations, the population is uncontrolled and follows Eq. (1) for the Ricker map f ðxÞ ¼ x expðrð1�x=KÞÞ with growth parameter r¼3 and carryingcapacity K¼60. In the next 20 generations, the population is controlled by adaptive threshold harvesting with intensity c¼ 2=3.

d

( )f d

1 /t tx x c+ =

1 ( )t tx f x+ =

1t tx x+ =

KTA /TA c tx

1tx +

Fig. 2. Adaptive threshold harvesting only takes place when the straight line atþ1

¼ at=c is under the graph of the population production map f. The activationthreshold AT is defined by their intersection. The bold red curve represents thepopulation production function (3) for the controlled system. (For interpretation ofthe references to color in this figure caption, the reader is referred to the webversion of this paper.)

J. Segura et al. / Journal of Theoretical Biology 395 (2016) 103–114 105

e.g. the Ricker (1954), Hassell (1975), and generalized Beverton–Holt (Bellows, 1981) maps.

2.2. Modelling adaptive threshold harvesting

Adaptive threshold harvesting exerts control on a populationwhen the population size xt at time step t exceeds a certain pro-portion of its value in the preceding generation. The control thenrestores the population size back to that threshold by harvestingthe surplus individuals. Thus, ATH takes action if the populationhas grown beyond a certain proportion within a time step, whichis why ATH can be seen as aiming to prevent population booms.

As ALC, this new limiter method is “adaptive” because themagnitude of the intervention is nonconstant and depends on thesystem state at the previous time step. Fig. 1 shows how ATHmodifies the dynamics of the population and, in particular, howthe fluctuation range of the population size is reduced.

When applying ATH, there are two different population sizes attime step t, namely bt, the population size before ATH interventionand at, the population size after intervention. In particular, btZatbecause ATH never adds individuals to the population. With thesenotations, the dynamics of ATH is determined by the followingsystem of difference equations:

btþ1 ¼ f ðatÞ and atþ1 ¼btþ1; btþ1rat=c;

at=c; btþ14at=c;

(ð2Þ

where cAð0;1Þ is a control parameter measuring the ATH inten-sity. Substituting the value for btþ1 from the first equation ofsystem (2) into the second one, we obtain that the populationdynamics is determined by the first-order difference equation

atþ1 ¼f ðatÞ; f ðatÞrat=c;

at=c; f ðatÞ4at=c;

(ð3Þ

which is piecewise smooth and can be written in one line by usingthe minimum function,

atþ1 ¼minff ðatÞ; at=cg:

2.3. Activation threshold

By definition, ATH only takes effect when the population sizeexceeds a proportion of its magnitude in the preceding generation.Any management implementing control must therefore wait untilmeasuring the population size in generation t to decide about theneed for intervention. Fortunately, the analysis of (3) reveals theexistence of a ‘hidden’ threshold level AT such that no control willbe necessary in generation t if the population size in the precedinggeneration is above this value.

Such knowledge can be of practical interest because no pre-paration will need to be taken in this situation. Note that the

activation threshold for ALC has a different meaning, namely thatcontrol actions only take place if the activation threshold isexceeded in the preceding time step. In order to calculate theactivation threshold for a given control parameter, we need toknow the population production function which can be obtainedfrom fitting to time series data. Proposition 1 in the Appendixshows the existence of AT for control intensities c4 infxA ð0;bÞ x=f ðxÞand its precise definition.

Although the determination of AT in practical situations can bedifficult due to noise or lack of information about the systemvariables, in deterministic systems its knowledge helps the con-troller to know early on when an intervention is unnecessary inthe next generation.

3. Stabilization of fluctuations

Here we describe how adaptive threshold harvesting affectsconstancy stability, i.e. the propensity of the population size tostay essentially unchanged (Grimm and Wissel, 1997). First of all,ATH is not able to stabilize oscillations towards an equilibriumpoint (see Proposition 2 in the Appendix). This is a property thatATH shares with ALC (Franco and Hilker, 2013). In the remainder ofthis section, we consider two different measures of constancystability, namely the fluctuation range and the fluctuation index.

J. Segura et al. / Journal of Theoretical Biology 395 (2016) 103–114106

3.1. Fluctuation range

The fluctuation range gives the upper and lower bounds of thepopulation size, in between which the oscillations take place. Ithas been employed by Franco and Hilker (2013) to study stabilityproperties of ALC. The smaller the fluctuation range, the morestable the population dynamics from the constancy point of view.

The ATH method can reduce the fluctuation range compared tothe uncontrolled system. We illustrate this in Fig. 3a, where thefluctuation range decreases as the control intensity is increased.The figure suggests that ATH confines the population size within aregion around the positive but unstable equilibrium K. The fol-lowing theorem states that such a “trapping region” indeed existsand is completely determined by the map f and the control para-meter c. The proof is provided in the Appendix.

Theorem 1. Assume that (C1)–(C3) hold and cA ð0;1Þ is such thatthe activation threshold AT exists. Then, applying ATH with intensity cconfines the population sizes at for any a0Að0; bÞ within an intervalIa ¼ ½lðcÞ;uðcÞ� around the positive equilibrium K, with endpointsgiven by the expressions

lðcÞ ¼f ðAT=cÞ; drAT ;

f ðf ðdÞÞ; d4AT ;

(and uðcÞ ¼

AT=c; drAT ;

f ðdÞ; d4AT :

(ð4Þ

If a specific goal is to be achieved, such as suppressing thepopulation size below an upper limit; beyond a lower limit; orwithin two limits, the control intensity to achieve these goals canbe determined. This is possible because the trapping region iscompletely determined by the map f and the control parameter.

The trapping region given in Theorem 1 is global, that is, thereduction of the fluctuation range does not depend on the initialcondition. Fig. 3a shows a bifurcation diagram for ATH togetherwith the limits of the intervals defining the trapping region givenby Eq. (4). These intervals are sharp over a wide range of controlparameters, which means that they cannot be improved for thoseparameter values.

ATH and ALC reduce the fluctuation range in different ways forthe considered Ricker map. On the one hand, ALC asymptoticallyprovides, for almost all control intensities, a lower limit for thepopulation size that is clearly higher than the one observed for the

Fig. 3. (a) Bifurcation diagram for ATH. Red dots represent population sizes for the systemby Eq. (4). (b) Bifurcation diagram for ALC with blue dots representing asymptotic populabehavior for c41 and consider the correct underlying equation atþ1 ¼maxff ðat Þ; c � atglimits of the fluctuation range for the uncontrolled system. The diagrams are based on thThe initial population size is chosen as a pseudorandom number in ð0; f ðdÞ�. The diagramof the references to color in this figure caption, the reader is referred to the web versio

uncontrolled system. The upper limit is significatively reduced forhigh intensities only, see Fig. 3b. On the other hand, ATH reducesthe upper limit in comparison to the uncontrolled system over awide range of control intensities, see Fig. 3a. The lower limit,however, increases significantly only for high control intensities.

Fig. 3a also shows the case when the control intensity c isgreater than unity. This corresponds to harvesting a populationthat has decreased below rather than increased above a fraction ofits previous size. Such a choice of control leads to extinction. Thereason is that the harvesting always forces the population size to afraction 1=co1; as a consequence, there is no positive fixed point.Choosing a control intensity c41 for ALC leads the population toblow up (Fig. 3b). This is because the control forces the number ofindividuals to increase in each generation to at least the propor-tion c41 of its previous value.

Finally, we point out that we do not include the special casec¼1 in our analysis. In this case, the asymptotic behavior dependson the initial condition for both ATH and ALC, since there is acontinuum of nontrivial equilibria—namely any population sizegreater than or equal to K, for ALC, or any population size less thanor equal to K, for ATH. This has not been reported before.

3.2. Fluctuation index

In Fig. 3, we have seen that constancy stability measured interms of the fluctuation range is always enhanced when ATH (orALC) is applied to the Ricker model considered in this paper.Looking at another measure of constancy stability, namely thefluctuation index (FI), we will show that ATH does not always havea stabilizing effect.

The fluctuation index is a dimensionless measure of the aver-age one-step variation of the population size scaled by the averagepopulation size in a certain period. It was introduced in Dey andJoshi (2006), and employed by Sah et al. (2013) and Franco andHilker (2013) to study stability properties of ALC.

Mathematically, the FI is given by

FI¼ 1Ta

XT�1

t ¼ 0

atþ1�at�� ��; ð5Þ

(3) controlled by ATH. The bold black curves mark the limits of the intervals givention sizes. This diagram is analogous to Sah et al. (2013), except that we include the, cf. Franco and Hilker (2013). In both panels, the horizontal dashed lines mark thee Ricker map f ðxÞ ¼ x expðrð1�x=KÞÞ with r¼3 and K¼60, after removing transients.s do not include the behavior at c¼1, cf. the main text for details. (For interpretationn of this paper.)

Fig. 4. Fluctuation indices (FIs) for the system controlled by ATH (red squares) andthe system controlled by ALC (blue dots). The horizontal line marks the FI for theuncontrolled population. The dynamics of the uncontrolled population is describedby the Ricker map f ðxÞ ¼ x expðrð1�x=KÞÞ with r¼3 and K¼60. The initial popu-lation size is chosen as a pseudorandom number in ð0; f ðdÞ�, and the FI is obtainedover 1000 generations after removing transients. (For interpretation of the refer-ences to color in this figure caption, the reader is referred to the web version of thispaper.)

0.2 0.4 0.6 0.8 1.0

20

40

60

80

Meanyieldpergeneration

0

Control parameter, cFig. 5. Mean yield per generation obtained by adaptive threshold harvesting. Thedashed curve represents the asymptotic yield after discarding transients, and thesolid curve represents the short-term yield including transients. Both yield valuesare averaged over 50 generations and over 1000 replicates with different initialconditions. The dynamics of the uncontrolled population are described by theRicker map f ðxÞ ¼ x expðrð1�x=KÞÞ with r¼3 and K¼60. The initial population sizeis chosen as a pseudorandom number in ð0; f ðdÞ� for the transient yield, and in thetrapping region Ia for the asymptotic yield.

J. Segura et al. / Journal of Theoretical Biology 395 (2016) 103–114 107

where a is the mean population size over a period of T time steps.Fig. 4 shows that for small values of the control parameter, the

FI of the system controlled by ATH can be greater than the FI of theuncontrolled system. The same holds true for ALC (Sah et al.,2013). For small values of the control intensity, the FI behavesquite erratically for both ATH and ALC, with pronounced peaks aswell as sporadic drops below the baseline level set by theuncontrolled system. These sudden changes are due to attractortransitions, e.g. from chaos to periodic oscillations or betweencycles of different periods (cf. Fig. 3).

For medium and high control intensities both ATH and ALCreduce the FI compared to the uncontrolled system. With a fixedvalue of the control intensity in this range, the effects of ATH andALC on the FI are not only qualitatively but also quantitativelysimilar.

4. Yield

Control strategies usually come at a price. Previous papersmeasure this cost in terms of the “effort”, i.e. the number ofindividuals added to or removed from a population (Hilker andWesterhoff, 2005; Dattani et al., 2011; Sah et al., 2013; Franco andHilker, 2013; Tung et al., 2014). Here, we take a slightly differentviewpoint and interpret the number of individuals removed as theyield. This seems to suggest itself since ATH is a harvestingmethod, provided that the population controlled is of some value.If the population is a pest, however, the term effort may be morefitting. In any case, we will consider two different ways of calcu-lating the yield; one is based on the long-term dynamics andignores transient effects (asymptotic yield), while the other isbased on the short-term dynamics only and thus takes intoaccount transients (transient yield). The reason for consideringtwo measures of the yield is the following. For ALC, on the onehand, the asymptotic effort has been shown to decrease to zero ifthe control intensity approaches its maximum value (Sah et al.,2013). On the other hand, including transients can radically alterthis observation and make the effort increase drastically (Francoand Hilker, 2013).

Fig. 5 shows the mean asymptotic yield per generation as afunction of the ATH intensity. The curve has a hump-shaped form,reaching a maximum at some intermediate value of the controlintensity. For too small control intensities, the yield is zero butthen increases with c until the maximum is reached. For largecontrol intensities beyond the maximum, the yield declines to zeroas c-1.

Fig. 5 also shows the mean transient yield per generation,which is similar to the asymptotic yield for small and intermediatecontrol intensities and reaches a local maximum, which is thesame as for the asymptotic yield (Fig. 5). For large control inten-sities (c40:85), however, the transient yield increases steeply,blowing up as c-1.

The reason for this sharp increase in the yield are prolongedtransients of the population dynamics if the control intensity islarge. This has the effect that the population is repeatedly har-vested—and, as a matter of fact, to a large degree. This not onlyincreases the yield, but also extends the time it takes the system toreach the trapping region. The following section considers this inmore detail.

5. Short-term behavior

In the previous section, we have seen that, in the short term,the mean yield per generation can differ greatly from the one inthe long term. In certain circumstances, there is a long transientperiod before the population reaches its asymptotic behavior.During this transient, the short-term dynamics can be markedlydifferent from the long-term dynamics with the desired properties(e.g., the mean yield or population size). Here, we show that ATHinduces prolonged transient periods for rather large values of thecontrol parameter. In the second part of this section, we proposean adjustment to the control rules that suppress prolonged tran-sients and accelerate the controlled population to reach its long-term dynamics.

5.1. Transients in the controlled system

In order to quantify transients, we consider a value tmax thatrepresents the maximum number of generations needed for the

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

Control parameter, c

Maximum

transient,tmax

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

Control parameter, c

Maximum

transient,tmax

Fig. 6. (a) Maximum transients for ATH in solid red curve and for ALC in dashed blue curve. (b) Maximum transients for the adjusted version of ATH given by Eq. (6) in solidred curve and for the adjusted version of ALC given by Eq. (7) in dashed blue curve. The value tmax represents the maximum transient among all orbits with integer initialpopulation size in the interval ð0; f ðdÞ�. The dynamics of the uncontrolled population are described by the Ricker map f ðxÞ ¼ x expðrð1�x=KÞÞ with K¼60 and r¼3. (Forinterpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

J. Segura et al. / Journal of Theoretical Biology 395 (2016) 103–114108

population size to enter the trapping region, measured over allpossible integer initial conditions in the interval ð0; f ðdÞ�. By takingthe maximum, we take into account that transients depend on theinitial condition, and we consider the ‘worst case’, i.e. the longesttime it takes to reach the trapping region.

Fig. 6a shows that the maximum transient increases with thecontrol intensity for both ATH and ALC. This increase is such thatthe maximum transients blow up near the maximum value of thecontrol intensity, c¼1. For medium and high control intensitiesATH exhibits a longer maximum transient than ALC.

Example 1. For the Ricker map f ðxÞ ¼ x expðrð1�x=KÞÞ with K¼60and r¼3 the maximum transient tmax for ALC with intensityc¼0.95 is 31 generations, whereas the corresponding value forATH with the same intensity rises up to 134 generations.

Analyzing the causes of this behavior can help us to devisetechniques for the cancellation of transients induced by ATH.Numerical simulations show that the maximum transients for ATHcorrespond to the largest values of the initial population size, a0.The reason is that, due to the overcompensation in the populationdynamics, the value a1 ¼ f ða0Þ is very small for large values of a0(see Fig. 2); the population size must then increase until enteringthe trapping region. However, the control slows down the popu-lation growth. In fact, for a sufficiently large value of a0, we havea1oAT . That is, harvesting control will be triggered in the next andsuccessive generations during this transient period. In summary,the prolonged transients are due to repeated high-intensity har-vesting of small population sizes. Since small population sizes canhave a large production, the repeated high-intensity harvesting isalso the reason why the transient yield increases sharply for largecontrol intensities (cf. Fig. 5).

5.2. How to reduce transients

As the transient dynamics may last for a long time, an impor-tant question from the practical point of view is whether thesystem can be manipulated to reach its long-term behavior faster.One of the simplest solutions is probably to apply a perturbationsuch that the population size directly enters the trapping region.However, depending on the current population size, this requiresthat both restocking and culling can be implemented promptlyand to a possibly large extent. Here we consider the situation thatrestocking is impossible (or very costly), so that, corresponding to

ATH, culling is the only possibility. We will propose an adjustedATH method that cancels the prolonged transients without chan-ging the asymptotic dynamics.

In the first part of this section, we have seen that the longtransients are caused by repeated harvesting at small populationsizes. The higher the control intensity, the more the populationgrowth is slowed down and the longer it takes the population sizeto reach the trapping region. This effect can be avoided by thefollowing two considerations.

1. We could stop harvesting small populations, say when thepopulation size is below the trapping region. The lower limit ofthe trapping region is l(c), but since the activation threshold AT

is always inside the trapping region, it would be actually suffi-cient to stop harvesting when the population size is smallerthan c � lðcÞo lðcÞ. For any at4c � lðcÞ, the population size in thenext generation is atþ1 ¼ at=c4 lðcÞ.

2. However, a complete cessation of the harvesting might causethe population size to “jump” from one side of the trappingregion to the other without entering it. Therefore, instead ofcompletely canceling the harvesting of the small populationsizes identified, we allow control but only to such a degree thatthe population size is not reduced below the trapping region.

All this leads to an adjusted ATH strategy with restricted harvesting atsmall population sizes, described by the following equation:

atþ1 ¼minff ðatÞ; at=cg; at4c � lðcÞ; ðoriginal ATH of large enough popns Þminff ðatÞ; lðcÞg; atrc � lðcÞ: ðrestricted harvesting of small popns Þ

(

ð6ÞThis adjustment does not alter the asymptotic behavior of the

controlled population because harvesting is only restricted whenthe population size is below the trapping region. Once it entersthis region, the control follows the original ATH. We illustrate theimprovement due to the adjustment (6) in Fig. 6b, which shows asignificant reduction in maximum transients.

Example 2. The maximum transient in the system from Example 1controlled by ATH with intensity c¼0.97 lasts for 134 generations.With the adjustment given by (6), this transient decreases to only4 generations.

The same kind of adjustment can be used to cancel prolongedtransients generated by ALC. In that case, the restocking action of

J. Segura et al. / Journal of Theoretical Biology 395 (2016) 103–114 109

the control lengthens transients when the population size is abovethe trapping region. For at4uðcÞ, where u(c) denotes the upperlimit of this region, the transient lasts until the population sizedecreases and enters the trapping region. Since the activationthreshold is always inside the trapping region, restocking takesplace in every generation of this stage, thus slowing down thedecrease of the population size. The higher the control intensity,the more the population is restocked and the longer it takes forthe population to decrease to the trapping region. As before, if therestocking control for population sizes above the trapping regionwere completely stopped, the population size could jump fromone side of the trapping region to the other without entering it.The proper adjustment is therefore to restrict restocking to theupper limit u(c). This leads to an adjusted version of ALC withrestricted restocking at large population sizes described by thesystem

atþ1 ¼maxff ðatÞ; c � atg; atoc � uðcÞ; ðoriginal ALC of small enough popns Þmaxff ðatÞ;uðcÞg; atZc � uðcÞ: ðrestricted restocking of large popns Þ

(

ð7ÞAs for ATH, this adjustment does not alter the asymptotic behaviorof the controlled population because the restocking is onlyreduced when the population size is above the trapping region.Once the system is in the trapping region, the control follows theoriginal ALC scheme.

Example 3. We consider the same uncontrolled population as inExamples 1 and 2. Applying ALC with restocking intensity c¼0.97 andfor initial conditions a0 that are integer values in ð0; f ðdÞ�, the max-imum transient lasts 31 generations. With the adjustment given by (7),the maximum transient decreases to only 4 generations.

6. Model extensions

So far, we have seen how ATH can reduce erratic fluctuations inthe population size for generic models with overcompensation. Inthis section, we extend the models to include different forms ofnoise, an Allee effect, and the lattice effect, i.e. integerization ofpopulation sizes. We will focus on the impact of these factors onthe constancy stability promoted by ATH.

One way to include all three model extensions is a stochasticdiscrete-state difference equation of the form

atþ1 ¼min f ðatÞIðatÞ exp σεt�σ2

2

� �� �; ⌈at=c⌉

� �; ð8Þ

where f(a) is the production function and the minimum-operatormodels ATH with intensity c as before. There are three modifica-tions. First, the factor I(a) brings in positive density dependence tomodel a strong Allee effect. Here we assume IðaÞ ¼ sa=ð1þsaÞ,which represents the probability of finding a mate, with s40measuring an individual's searching efficiency (Schreiber, 2003).Second, the flooring function ⌊ac maps a number to the greatestinteger smaller than or equal to a, whereas the ceiling function ⌈a⌉maps a number to the smallest integer greater than or equal to a.This integerization takes into account that individuals alwayscome and are harvested in whole numbers. Third, we include noisein the form of environmental stochasticity, where εt is a normallydistributed variable with expectation 0 and variance 1, and σ is aparameter measuring the noise intensity.

The bifurcation diagram in Fig. 7a shows that ATH reduces thefluctuations in population size in a similar way like the determi-nistic, continuous-state model without Allee effect. Apart from thenoisy appearance, the major difference is that too small popula-tions go extinct. The fluctuation index is shown in Fig. 7b for dif-ferent intensities of environmental stochasticities and different

strengths of Allee effects. ATH is effective in reducing the fluc-tuation indices for the extended model over a similar range ofcontrol parameters as for the model without the extensions.Interestingly, it appears that control is less counterproductive atsmall values of c.

Another major source of noise in ecological systems is demo-graphic stochasticity. This can be modeled in the following way(Brännström and Sumpter, 2006, cf.):

atþ1 ¼min f ðatÞIðatÞ expffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

σ2

f ðatÞIðatÞ

sεt�

σ2

2f ðatÞIðatÞ

( )$ %; ⌈at=c⌉

( );

ð9Þwhere all the variables have the same meaning as before. Theresults in terms of population fluctuations and fluctuation indices(Fig. 7c and d, respectively) are similar to the model with envir-onmental stochasticity.

To summarize, the stabilizing effect of ATH seems to be robustagainst the lattice effect, Allee effect and both environmental anddemographic noise, even if their intensity is high. Even though onemight expect that noise increases fluctuations, the fluctuationindex in the stochastic model with Allee and lattice effect issmaller than in the deterministic baseline model for some valuesof the control parameter. In this sense, the model extensions seembeneficial for stabilization.

For comparison, ALC has been shown to be robust against thelattice effect and noise as well (Sah et al., 2013). Allee effects werenot considered, and noise was modeled in form of uniformly dis-tributed random numbers added to the model parameter. How-ever, if the threshold value is measured before intervention (ALCb),noise and the lattice effect can produce counterproductive effectsof control that are related to multiple attractors (Franco and Hilker,2014).

7. Discussion and conclusions

The original version of adaptive limiter control (ALC) as pro-posed by Sah et al. (2013) is based on restocking the population ascontrol intervention. In this paper, we have shown that adaptivethreshold harvesting (ATH), i.e. the harvesting version of adaptivelimiter control (h-ALC), can also stabilize fluctuating populationsizes. This extends the applicability of adaptive limiters to situa-tions when culling is the only possible form of intervention.Moreover, adaptive limiters may also be used as a harvestingstrategy, thus widening the approach of threshold harvesting.

7.1. Commonalities between ATH and ALC

ATH and ALC have the following properties in common. Webegin by considering stabilization aspects. First, both controlstrategies confine the population size in an interval around theunstable fixed point of carrying capacity. This guarantees thatpopulation size fluctuations are bounded and means that boomsand busts of a population become restricted. We provide analyticalexpressions for the lower and upper bounds, which correspond tominimum and maximum population sizes during the oscillations.We prove that, for sufficiently large control intensity, the range ofpopulation size fluctuation reduces with increased control inten-sity, meaning that the control strategy is effective with respect toreducing the extent of population cycle amplitudes. Hence, con-stancy stability tends to improve.

Second, the fluctuation index also becomes smaller withincreased control intensity, provided the control is at least ofintermediate strength. Hence, constancy stability improves. How-ever, for smaller control intensities both ATH and ALC may

J. Segura et al. / Journal of Theoretical Biology 395 (2016) 103–114110

increase the fluctuation index compared to the uncontrolledpopulation. Third, the carrying capacity itself does not becomestabilized. It is only the fluctuation in population size, as measuredby the fluctuation index or the range of possible population sizesfrom the peak to the trough, that reduces in magnitude oramplitude. Yet, for the Ricker model studied in this paper, themean population size is, asymptotically, remarkably constant atthe level of the carrying capacity, for all values of the controlparameter 0oco1.

Hence, the stabilization properties of ATH are analogous to theones of ALC (previously investigated in Sah et al., 2013; Franco andHilker, 2013). This is not a straight-forward result; as pointed outin the introduction, there are other control strategies which cangreatly differ in the behavior they trigger, depending on whetherinterventions either add or remove individuals.

Moreover, the length of transients and the yield (or effort,respectively) behave similarly as a function of the control intensityfor ATH and ALC (for the latter, again, this has been previouslyinvestigated in Sah et al., 2013; Franco and Hilker, 2013).

Fig. 7. Stabilization of population size fluctuations achieved by ATH in integerized stochbased on Eq. (8). Bottom row: demographic stochasticity, based on Eq. (9). The left columstrength s¼0.15 after removing transients. Initial population sizes are pseudorandom nshows the fluctuation index as a function of the control parameter. The FI has been comreplicates. The production function is the Ricker model with r¼3 and K¼60.

7.2. Differences between ATH and ALC

While both ATH and ALC reduce the fluctuation range for theRicker model studied in this paper, they affect the lower and upperbounds of population size fluctuation in different ways. Thismeans that the peaks and troughs possible during populationsfluctuations respond differently to a control strategy that isimplemented more intensively. For intermediate control inten-sities (cA ½0:4;0:7�), ATH tends to reduce the upper bound in amore pronounced way, whereas ALC tends to increase the lowerbound more markedly (see Fig. 3). Hence, ATH appears moreeffective in avoiding outbreaks and ALC in preventing extinctions.For large control intensities (cAð0:7;1Þ), however, both controlmethods seem equally effective in assuring minimum and max-imum population sizes. We ought to mention that population sizeis here measured after control. If the population were censusedbefore control, the interventions could affect the fluctuation rangeand fluctuation index differently, as shown in Franco and Hilker(2013, 2014) for ALC.

There are other differences between ATH and ALC. They occurwhen the control intensities exceed values of unity. Note that thisparameter range has not been considered before for ALC. Basically,

astic models including a strong Allee effect. Top row: environmental stochasticity,n shows asymptotic bifurcation diagrams for noise intensity σ ¼ 0:05 and Allee effectumbers in ð0;5M=2�, where M denotes the maximum of f ðxÞIðxÞ. The right columnputed over 100 iterations (after rejecting 900 transients) and is averaged over 100

J. Segura et al. / Journal of Theoretical Biology 395 (2016) 103–114 111

c41 means restocking the population to levels larger than beforea crash (ALC) and harvesting the population to a level smaller thanbefore the increase in population size (ATH). As such, these areneither unrealistic parameter regimes nor senseless strategies.However, we find that these high control intensities drive thepopulation extinct (ATH) or lead to unbounded population growth(ALC). Neither of which seems a desirable goal for the strategyconsidered.

We point out that the dynamics inside the trapping region arenot theoretically analyzed in this study. Such an analysis seems aninteresting line of research for which mathematical tools fromErgodic Theory could be useful.

7.3. Comparison with threshold harvesting

In contrast to adaptive limiters, threshold harvesting canactually stabilize the population dynamics to a stable equilibriumfor sufficiently large control values (Glass and Zeng, 1994; Sinha,1994). However, this comes at the cost of applying the interven-tion in every generation.

The other major difference is that, by definition, thresholdharvesting affects only large population sizes. The long-term meanyield per generation tends to be a multimodal function of thethreshold (cf. Fig. 3.9c in Hilker and Westerhoff, 2005). ATH, bycontrast, is a harvesting policy that by its design is relevant also forpopulations with smaller sizes, provided they have grown suffi-ciently in comparison to the previous census. Interestingly, ATHshows a unimodal pattern in the long-term mean yield per gen-eration (Fig. 5).

7.4. Yields and transients

While the primary goal of ATH is the stabilization of the fluc-tuations, it may also be applied as a harvesting strategy when theaim is to gain economic benefit from the exploited population.ATH may therefore represent an alternative to other harvestingstrategies such as constant-effort, constant-yield, or thresholdharvesting.

It turns out that we have to distinguish two situations, namelythe long-term (asymptotic) and the short-term (transient) yield.Interestingly, the short-term yield gained by ATH rises sharply forvalues of high control intensities. That is, the yield becomes largestjust before the population goes extinct for c41. The transitionfrom a sustained population (with improved constancy stabilityand large yield) to extinction happens abruptly. In contrast tooverexploitation with constant-effort harvesting, the collapse ofthe population does not take place gradually.

Trying to maximize the short-term gain is therefore risky interms of sustainability. This bears some analogy to the observationthat focusing on short-term gains can lead to dramatic con-sequences. One of the most prominent examples is probably thecollapse of the cod stocks off of Newfoundland. Harvesting theoryhas therefore developed strategies for a sustainable catch, whichcan be considered as one of the cornerstones of mathematicalbioeconomics (Clark, 1990). In fisheries, particular attention is paidto the maximum sustainable yield (MSY). Even though there aremany concerns regarding the concept of MSY (e.g. Larkin, 1977;Ludwig et al., 1993), it remains a “key paradigm in fisheriesmanagement” (Maunder, 2008, p. 2295).

As a consequence, the harvesting literature has focused almostexclusively on long-term behavior and asymptotic yields. This is incontrast to the realization that harvesting represents additionalperturbations to the population, and that the population rarelyreaches its equilibrium state (Fox and Gurevitch, 2000). Whiletransient dynamics are well-known to be important (Hastings,2004), there is little work that aims to optimize harvest taking into

account transient regimes (Jensen, 1996; Hauser et al., 2006;Koons et al., 2006, 2007). In stochastic population models, wherepopulation extinction is inevitable in the long run, Lande et al.(2003, p. 122) addressed this by considering the “expectedcumulative yield over all time before eventual extinction of thepopulation or reduction to a specific size”. However, the time tostochastic population extinction may be quite long.

In this paper, we have shown that the yield can be markedlydifferent, depending on whether we consider a short or long timescale. The dramatic increase in the short-term yield for largecontrol intensities can be readily explained by the time it takes thepopulation to reach the trapping region. During this time, thepopulation is always harvested; the intervention frequencyapproaches 1 (not shown). That is, during this time the populationeffectively follows atþ1 ¼ at=c with c close to one, which corre-sponds to geometric growth with a rather slow per-capita pro-duction. Hence, due to the high yield from harvesting the popu-lation growth is significantly reduced and almost stopped.

Here, we have calculated the transient yield over a time hor-izon of 50 generations. This is arbitrary and could be varied.

7.5. Dealing with transients

The long transients that occur for high control intensities mayappear quite desirable on the one hand because they incrementthe transient yield and simultaneously reduce the asymptoticfluctuation range. On the other hand, however, the prolongedtransient period keeps the population size at low levels and pre-vents it from reaching the trapping region. In many practicalsituations, e.g. when it comes to supporting endangered species, itis imperative to reduce the transients.

A major drawback of adaptive limiters is that they triggercontrol actions whenever the population size exceeds (or fallsbelow) a proportion of its magnitude in the preceding generation—regardless of whether this magnitude is close to zero (or on ahigh level, respectively). This can happen when the controlintensity in both ATH and ALC is large. Yet, it does not seem tomake sense to harvest a population that has increased in sizewhen this size is still small and far below the trapping region.Similarly, for ALC, it seems unreasonable to restock a populationthat has declined but is still above the trapping region.

Based on this observation, we propose adjustments to bothATH and ALC that suppress prolonged transients, while retainingthe asymptotic behavior. These adjustments concern only popu-lation sizes outside the trapping region and are such that thepopulation size in the next generation does not ‘overshoot’ or‘undershoot’ the trapping region. This is achieved by restrictingthe harvesting or restocking intervention for ATH and ALC,respectively. If the population size is within the trapping region,there is no need to alter the original control schemes because thepopulation does not leave the trapping region. The adjustmentswork well (Fig. 6a, Examples 2 and 3) and are therefore effective inspeeding up the transition from a transient period to the asymp-totic regime.

As mentioned previously, transients are rarely taken intoaccount or studied (but see Labra et al., 2003; Caswell, 2007; Ellis,2013; Franco and Hilker, 2013; Franco and Ruiz-Herrera, 2015),even though they are ubiquitous in nature and may be actuallymore important than long-term dynamics (e.g. Hastings, 2004).The problem of directing an unstable or perturbed population inan efficient way to a desired state, such as the equilibrium, bearssome analogy to the idea of targeting in chaos control (Kostelich,1999; Hilker and Westerhoff, 2007b). In the ecological literature,we could not find many studies that investigate how to deal withtransients. Harley and Manson (1981) suggested an “intermediateharvesting policy” for the transient period that accelerated the

J. Segura et al. / Journal of Theoretical Biology 395 (2016) 103–114112

transition to the equilibrium state of a structured population.Another, yet completely different approach is based on utilizingavailable time series; learning from ‘trajectories from the past’ onecould steer the system to a desired state efficiently (Hilker andWesterhoff, 2007a,b).

This time-series-based approach has the advantage of notrequiring any knowledge of the underlying laws of dynamics. Aspointed out by Sah et al. (2013) for ALC, one of the main advan-tages of adaptive limiters over other strategies for controllingbiological populations is that they can be implemented even whena good estimation of the population production map f for theuncontrolled system is not available. In this situation, the lack ofknowledge about the system behavior makes it very difficult toreduce the length of transients. The adjusted methods presentedhere do require information on the upper or lower bounds of thetrapping region.

Control strategies that are aimed at biological populations andthat efficiently dealt with transient while requiring little infor-mation seem therefore an interesting research endeavor.

Acknowledgements

D. Franco was partially supported by the ETSI Industriales(UNED), project 2015-MAT07, and by the Ministerio de Economía yCompetitividad and FEDER, Grant MTM2013-43404-P.

Appendix A. Proofs of the analytical results

Throughout this appendix, f t refers to the tth iterate of f, i.e.,f t ¼ f○f t�1.

Proposition 1. Assume that (C1)–(C3) hold. For control intensitiesc4 infxA ð0;bÞ x=f ðxÞ the set

W ¼ xAð0; b� : f ðxÞ ¼ x=c�

is nonempty and has a maximum AT oK . If cr infxA ð0;bÞ x=f ðxÞ theaction of ATH is never triggered.

Proof. Consider the functions gðxÞ ¼ x=f ðxÞ and hðxÞ ¼ f ðxÞ�x=c. By(C2) we have gðxÞo1 in ð0;KÞ and gðxÞZ1 in ½K ; bÞ, therefore theinfimum of g can only correspond to its image at a point in ð0;KÞ(in that case, a minimum) or to its one-side limit on the right of 0.Assume that there exists zAð0;KÞ such that infxA ð0;bÞ gðxÞ ¼gðzÞ ¼ z=f ðzÞ. Then, c4z=f ðzÞ and therefore hðzÞ ¼ f ðzÞ�z=c40. Onthe other hand, f ðKÞ ¼ KoK=c, and then hðKÞ ¼ f ðKÞ�K=co0.According to Bolzano's Theorem, there must exist a point ~x40with hð ~xÞ ¼ 0, and hence ~xAW . Now we consider the case forwhich infxA ð0;bÞ gðxÞ ¼ limx-0þ gðxÞ. Since f ð0Þ ¼ 0, the conditionfor the control parameter can be restated as

1co lim

x-0þ

1gðxÞ ¼ lim

x-0þ

f ðxÞ� f ð0Þx

¼ f 0ð0þ Þ:

By the linear approximation of f at x¼0 there exists a neighbor-hood U �Rþ with f ðxÞ � f 0ð0þ Þ � x4x=c for all xAU. Therefore,hðxÞ ¼ f ðxÞ�x=c40 for all xAU, and applying the same argumentas before we can conclude that there must exist a point ~x40 withhð ~xÞ ¼ 0, and hence ~xAW . This completes the proof that W isnonempty for the considered control intensities.

Since f is strictly decreasing for x4d and f ðKÞ ¼ K , we have thatW � ð0;KÞ. Moreover, the set h�1ðf0gÞ � ½0; b� is closed, and there-fore W ¼ h�1ðf0gÞ \ ð0; b�a∅ has a maximum AT oK .

Finally, for cr infxA ð0;bÞ x=f ðxÞ it is f ðxÞrx=c for all xA ½0; b� andtherefore minff ðxÞ; x=cg ¼ f ðxÞ. This means that the action of ATH isnever triggered for these intensities.□

Corollary 1. Assume that (C1)–(C3) hold and cAð0;1Þ is such thatthe activation threshold AT exists. Then the map describing thedynamics of at for the controlled system under ATH with intensity c,

HðxÞ ¼minff ðxÞ; x=cg; ðA:1Þverifies HðxÞ ¼ f ðxÞ for all xZAT and xrHðxÞrx=c for xrAT .

Proof. Since AT is the highest intersection point between f and thestraight line y¼x/c, the inequality f ðxÞrx=c must hold for allxZAT . Hence, HðxÞ ¼minff ðxÞ; x=cg ¼ f ðxÞ for these points. It isclear that HðxÞ ¼minff ðxÞ; x=cgrx=c. Since AT oK and f ðxÞ4x forall xoK , then f ðxÞ4x for all xrAT oK . Given that co1 it is x=cZx for all xZ0, and hence HðxÞ ¼minff ðxÞ; x=cgZx for all xrAT .□

Corollary 2. Assume that (C1)–(C3) hold and cAð0;1Þ is such thatthe activation threshold AT exists. Then ATH does not act in generationt if at�14AT .

Proposition 2. Assume that (C1)–(C3) hold and that the fixed pointK is unstable for the uncontrolled system (1). Then, independent ofthe magnitude of ATH, cAð0;1Þ, the controlled system has noasymptotically stable equilibria.

Proof. For cr infxA ð0;bÞ x=f ðxÞ the control by ATH is never trig-gered and the dynamics of the uncontrolled and controlled sys-tems are the same. By condition (C2), this system has only twofixed points x¼0 and x¼K. Since f ðxÞ4x for 0oxoK , x¼0 is anunstable equilibrium. Under the assumption that K is alsounstable, we conclude that the “controlled” system has noasymptotically stable equilibria.

For control intensities c4 infxA ð0;bÞ x=f ðxÞ the dynamics of thecontrolled system by ATH are different from the ones for theuncontrolled system and are strictly given by (A.1). Looking at thismap, it becomes clear that x¼0 is a fixed point. Since co1, wehave that fixed points with x40 must verify x¼ f ðxÞ, and thusx¼K. Hence, the system controlled by ATH with intensity c4infxA ð0;bÞ x=f ðxÞ has only two fixed points, namely 0 and K. We aregoing to prove that none of them is asymptotically stable. Considerthe neighborhood U ¼ ð0;AT Þ of 0. We are going to prove that allorbits starting in U eventually leave it. Assume atAU for all tZ0.By Corollary 1 it is HðxÞ4x for xAð0;AT Þ, which means atþ14at .Hence, the sequence ðatÞtAN is increasing and upper bounded, so itconverges to some point x⋆Að0;AT �. This point is a fixed point ofthe system, which is absurd because there is no fixed point in theinterval ð0;AT � � ð0;KÞ. Hence, we conclude that 0 is an unstablefixed point.

Let us now prove that K is also unstable. As f is continuous andf ðKÞ ¼ KoK=c, there exists a neighborhood V of K such that f ðxÞox=c for all xAV . Assume atAV for all tZ0. According to (A.1) itis atþ1 ¼ f ðatÞ for all tZ0, and thus at ¼ f tða0Þ. Since K is anunstable fixed point for the uncontrolled system, this last equalitycontradicts our hypothesis and allows us to conclude that K is anunstable fixed point for the controlled system by ATH.

Theorem 2. Assume that (C1)–(C3) hold and cAð0;1Þ is such thatthe activation threshold AT exists. Then, applying ATH with intensity cconfines the population sizes at for any a0A ð0; bÞ into an interval Ia¼ ½lðcÞ;uðcÞ� around the positive equilibrium K, with endpoints givenby the expressions

lðcÞ ¼f ðAT=cÞ; drAT ;

f ðf ðdÞÞ; d4AT ;

(and uðcÞ ¼

AT=c; drAT ;

f ðdÞ; d4AT :

(

Proof. In order to cover all possible expressions for Ia, we mustconsider two cases.

We start considering the case drAT , for which Ia ¼ ½f ðAT=cÞ;AT=c�. We have that drAT oAT=c, and thus f ðdÞZ f ðAT Þ4 f ðAT=cÞbecause co1 and f is strictly decreasing in (d,b). From this, weconclude that the interval Ia has nonempty interior because f ðAT Þ

J. Segura et al. / Journal of Theoretical Biology 395 (2016) 103–114 113

¼ AT=c and therefore AT=c4 f ðAT=cÞ. To prove that orbits enter Iaafter an initial transient, we consider exhaustive and disjoint casesdepending on the initial population size a0Að0; bÞ.1. Firstly, we assume that a0A ½AT ;AT=c�. In this case,

drAT ra0rAT=c;

and thanks to the strict decrease of f on (d,b) we conclude that

f ðdÞZ f ðAT ÞZ f ða0ÞZ f ðAT=cÞ:On the other hand, according to Corollary 1, it isa1 ¼Hða0Þ ¼ f ða0Þ. With this, given that f ðAT Þ ¼ AT=c, we havethat AT=cZa1Z f ðAT=cÞ and thus a1A Ia.

2. Next, we assume a0A ð0;AT Þ. We are going to show that thereexists t0AN such that at0 A ½AT ;AT=c�. Assume at =2½AT ;AT=c� forall t. Let us prove by induction on t that atA ð0;AT Þ for all t. Fort¼0 this condition is straightforward from the hypothesis of thecase. Suppose atAð0;AT Þ for certain tZ1. Since atoAT , we havethat atþ1 ¼HðatÞrat=c by Corollary 1, and therefore

atþ1rat=coAT=c:

As atþ1 =2½AT ;AT=c� by hypothesis, we conclude that atþ1Að0;AT Þ. In conclusion, atAð0;AT Þ for all t. By Corollary 1 it is atþ1

¼HðatÞ4at for all t, and hence the sequence ðatÞtAN is increas-ing and upper bounded by AT, so it converges to some pointx⋆Að0;AT �. This point must be a fixed point for the system, whatis absurd because there is no fixed point in the intervalð0;AT � � ð0;KÞ. In summary, we conclude that there exists t0AN such that at0 A ½AT ;AT=c� and thus at0 þ1A Ia by the firstsubcase.

3. Finally, we assume a0AðAT=c; bÞ. We have that x04AT=c4AT Zd and according to Corollary 1, a1 ¼Hða0Þ ¼ f ða0Þ. From the strictdecrease of f on (d,b) we conclude that

a1 ¼ f ða0Þo f ðAT=cÞo f ðAT Þ ¼ AT=c;

what leads to one of the previous subcases and proves thatorbits eventually enter Ia.

So far, we have proved that orbits enter the trapping region after afinite number of generations. To prove that they never leave it, wemust see that atþ1A Ia for atA Ia. Assume atA Ia for certain t. Iff ðAT=cÞZAT we have that AT r f ðAT=cÞratrAT=c, and thereforeatþ1A Ia by the previous subcase 1. For f ðAT=cÞoAT we considertwo cases. If atoAT it is atoatþ1 ¼HðatÞoat=c (Corollary 1) andthus AT=c4at=c4atþ14atZ f ðAT=cÞ, what implies atþ1A Ia. If atZAT it is atA ½AT ;AT=c�, and therefore atþ1A Ia by the previoussubcase 1.

We consider now the case d4AT , for which Ia ¼ ½f 2ðdÞ; f ðdÞ�. Iff 2ðdÞ ¼ f ðdÞ then f ðdÞ4K is a fixed point for f, in contradiction withcondition (C2). Hence, Ia has nonempty interior and we can pro-ceed to prove that orbits enter this interval after a finite number ofgenerations. To do this, we must distinguish some cases as before.1. Firstly, we assume that a0A ½d; f ðdÞ�. As a0Zd4AT , it is a1 ¼H

ða0Þ ¼ f ða0Þ by Corollary 1. Given that f is strictly decreasing on(d,b), from the inequality dra0r f ðdÞ we deduce thatf 2ðdÞr f ða0Þ ¼ a1r f ðdÞ, what implies a1A Ia.

2. Next, we assume a0A ð0; dÞ. We are going to see that there existst0AN such that at0 A ½d; f ðdÞ�. Suppose at =2½d; f ðdÞ� for all t. Giventhat f reaches its absolute maximum at x¼d, for all xA ½0; b� wehave that HðxÞ ¼minff ðxÞ; x=cgr f ðxÞr f ðdÞ. Hence, at ¼Hðat�1Þr f ðdÞ for all t, what together with at =2½d; f ðdÞ� leads to concludethat atA ð0; dÞ for all t. Since f ðxÞ4x for all xA ð0; dÞ � ð0;KÞ wehave that f ðatÞ4at for all t, and given that at=c4at it is atþ1 ¼HðatÞ ¼minff ðatÞ; at=cg4at . Hence, the sequence ðatÞtAN isincreasing and upper bounded by d, so it converges to somepoint x⋆Að0;d�. This point must be a fixed point for the system,what is absurd because there is no fixed point in the interval

ð0; d� � ð0;KÞ. We conclude that there exists t0AN such that at0A ½d; f ðdÞ� and, with this, at0 þ1A Ia by the previous subcase.

3. Finally, we assume a0A ðf ðdÞ; bÞ. In this case we have that 0oa1 ¼Hða0Þr f ðdÞ because HðxÞr f ðdÞ for all xA ð0; bÞ. This bringsus back to one of the previous cases and allows us to assert thatorbits eventually enter the trapping region.

Now we proceed to prove the invariance of the trapping regionunder ATH. To do this, we assume atA Ia for a certain t and proveatþ1A Ia. We have seen that atþ1 ¼HðatÞr f ðdÞ for all t, so it isenough to prove that atþ1Z f 2ðdÞ. For atZd this condition isstraightforward from the previous subcase 1. If atod we have thatf ðatÞ4at by condition (C2), and then

atþ1 ¼HðatÞ ¼minff ðatÞ; at=cgZminfat ; at=cg ¼ atZ f 2ðdÞ:□

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